@Kelly is correct! There are a few differences in those areas. For obstacle in AMHA, they must be 3yrs old vs AMHR where they can be yearlings.
They only use the showmanship patterns that are listed in their rulebook and they do use the quarter system vs the half system that AMHR uses.
And they do require closed wheels and reversing on the diagonal in roadster classes.
In addition, some classes have different names. Western Country becomes Classic Pleasure in AMHA for example. Jumping is decided by height rather than time. Horses are not encouraged to show stretched in halter. Western clothing is mandatory in Stock Type.
Other than that, I am not aware of any major differences on the local level offhand.
If you are at a local show that is A&R, the judges are usually the same for both. However, I have seen them use one or two different judges for an A show who were not in the R show, but they usually don't switch them all out. The show premium will give you that information.
The judging is the same as R at the local level but they use a different format for their Regional and World shows.
Just double check the rulebook and you shouldn't have any trouble!